Abstract
We calculate Zadeh’s max-min composition operators for two 3-dimensional triangular fuzzy numbers. We prove that if the 3-dimensional result is limited to 2 dimensions, it is the same as the 2-dimensional result, which is shown as a graph. Since a 3-dimensional graph cannot be drawn, the value of the membership function is expressed with color density. We cut a 3-dimensional triangular fuzzy number by a perpendicular plane passing a vertex, and consider the cut plane as a domain. The value of the membership function for each point on the cut plane is also expressed with color density. The graph expressing the value of the membership function, defined in the plane as a 3-dimensional graph using the z -axis value instead of expressing with color density, is consistent with the results in the 2-dimensional case.
Highlights
Many results exist for Zadeh’s max-min composition operators. e results for triangular fuzzy numbers are well known [1,2,3,4,5]
By limiting the graph of the 3-dimensional result to the 2-dimensional case, we prove that the result expressed as a graph is consistent with the graph of the 2-dimensional result
By limiting the graph of the 3-dimensional result to the 2-dimensional case, we prove that the result expressed as a graph is consistent with the graph of the 2dimensional result
Summary
Many results exist for Zadeh’s max-min composition operators. e results for triangular fuzzy numbers are well known [1,2,3,4,5]. Many results exist for Zadeh’s max-min composition operators. E results for triangular fuzzy numbers are well known [1,2,3,4,5]. We have generalized 1-dimensional triangular fuzzy number to 2-dimensional triangular fuzzy number and calculated Zadeh’s max-min composition operators for 2dimensional fuzzy numbers [6]. In the 1-dimensional case, Zadeh’s max-min composition operator can be calculated using α-cuts. By defining parametric operations between two region valued α-cuts, we obtained parametric operations for two triangular fuzzy numbers defined on R2 [7]. In the case of 3-dimensional fuzzy numbers, the α-cuts are subsets of R3. By defining parametric operations between two ellipsoids including interior valued α-cuts, we calculated Zadeh’s max-min composition operators for two 3-dimensional fuzzy numbers [8]. By limiting the graph of the 3-dimensional result to the 2-dimensional case, we prove that the result expressed as a graph is consistent with the graph of the 2-dimensional result
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